Question

1除以1*4+1除以4*7+1除以7*10-------+1除以（3N-2）*（3N+1）

Answer 1

1/1*4+1/4*7+1/7*10+............+1/(3n-2)(3n+1)
=1/3*(1-1/4)+1/3*(1/4-1/7)+1/3*(1/7-1/10)+...........+1/3*[1/(3n-2)-1/(3n+1)]
=1/3*[1-1/4+1/4-1/7+1/7-1/10+...........+1/(3n-2)-1/(3n+1)]
=1/3*[1-1/(3n+1)]
=1/3*[(3n+1-1)/(3n+1)]
=1/3*3n/(3n+1)
=n/(3n+1)

Answer 2

1除以1*4+1除以4*7+1除以7*10-------+1除以（3N-2）*（3N+1）
=1/1*4+1/4*7+1/7*10+......1/(3n-2)(3n+1)
=[(3/1*4)+3/4*7+3/7*10+.....3/(3n-2)(3n+1)]÷3
=[1-1/4+1/4-17/+1/7-1/10+1/10-.........-1/(3n-2)+1/(3n-2)-1/(3n+1)]÷3
=[1-1/(3n+1)÷3
=(3n+1-1)/3(3n+1)
=n/(3n+1)